*2.1. Individual TCL Characteristic*

The TCLs can consume electric energy and release thermal energy, which can usually be stored and transferred [18]. For example, the simple TCLs usually have two work states, i.e., "on" or "off", and each corresponds to one power value, i.e., *P*rate or 0. When there is excess power in generation side, the TCL is changed to the "on" state, then the electricity consumption increases and the transformed heat energy will also increase. When the power generation fails to meet users' needs, the TCLs are changed

to the "off" state and the power demand will be reduced in order to stabilize the power grid frequency. The characteristics are described as follows [19]:

$$\dot{T}(t) = \frac{1}{CR}(T\_d - T(t) - s(t)RP),\tag{1}$$

where *T* and *Ta* represent the indoor and outdoor temperature, respectively. *P*, *R*, and *C* denote energy transfer rate, thermal resistance, and thermal capacitance, respectively. *s*(*t*) represents the switching state of loads.

The operation characteristics of an individual TCL are shown in Figures 1 and 2. The set-point regulation method is adopted for regulating TCLs to achieve the purpose of peak shaving and load shifting. In the figure, *u* is the temperature set-point change. *T*+, *Ts*, and *T*− on the left of the figure denote the upper boundary of temperature, the temperature set-point, and the lower boundary of temperature, respectively. is the width of the temperature deadband, which denotes the difference between the upper and lower limits of temperature. According to the physical characteristics of TCLs, *T*+, *Ts*, *T*−, and have the following relationship:

$$\begin{cases} T\_+ = T\_s + \triangle / 2, \\ T\_- = T\_s - \triangle / 2, \\ \triangle = T\_+ - T\_-. \end{cases} \tag{2}$$

The rising edge represents that the room temperature has a rise caused by a natural heat conduction process when the TCL is off. The falling edge denotes the temperature drops caused by the cooling process of the TCL when the TCL is turned on. In order to keep the room temperature near the temperature set-point, the TCL will be changed from an off to an on state when the temperature reaches the upper limit and changed from on to off state when the temperature reaches the lower limit in Figure 1. We can observe that the upper and lower bounds of temperature vary with the temperature set-point, but the deadband keeps constant in Figure 2. Hence, when the temperature set-point changes, the TCL's switch state can be indirectly changed, and the TCL's running time will be extended or shortened. Therefore, prolonging or shortening the running time of TCL will change the demand-side power consumption, thus it is effective to maintain the stability of the power grid frequency.

**Figure 1.** Operation characteristic of an individual thermostatically controlled load (TCL) (*u* = 0).

Suppose that *N* TCLs are used to provide the frequency regulation. When *u* is changed, the power consumption in demand side will be shifted, which contributes to the peak shaving and valley filling for the grid. The aggregated power consumption *P*total could be expressed by

$$P\_{\text{total}}(t) = \sum\_{i=1}^{N} \frac{1}{\eta\_i} \mathbf{s}\_i P\_{i\star} \tag{3}$$

where *ηi* (>1) is the efficiency coefficient of the *i*th TCL. *si* is a binary variable. The TCL is on when *si* = 1; and off when *si* = 0. *Pi* denotes the rated power of the *i*th load.

**Figure 2.** Operation characteristic of an individual TCL (*u* = 0).
